Rahman Group Project Abstracts
Ising Model Simulation of a Two-Dimensional Antiferromagnetic Triangular Lattice
Sean Laird, 2006
No abstract available.
Computational Solution of the Antiferromagnetic Potts Model
Brandon Redding, 2005
This work discusses a series of topics in the Antiferromagnetic Potts model. Incorporating next-nearest neighbor interactions into the three-state model is discussed and implemented in terms of our energy calculations. However, there are difficulties involved when incorperatating these calculations into the Cluster-Flip Monte Carlo method used to study the model. Similar difficulties are discussed in relation to adding an external magnetic field component. The main work of this paper compiles spin distribution data for the 3, 4, 5, and 6 state Potts models. Studying these spin distributions can be useful when characterizing system phases. The behavior of the system is studied as the lattice size goes to infinity in order to clearly understand the validity of the model.
Five State Antiferromagnetic Potts Model: Monte Carlo Simulation
Joshua Monk, 2003
We examined the existence of phase transitions for a five-state antiferromagnetic Potts model on a simple cubic lattice using Monte Carlo simulation. We started by using the Metropolis algorithm for Monte Carlo simulation to examine simple systems in order to familiarize ourselves with standard ideas and techniques involved in such work. Later, we used the fast, highly efficient Swendsen and Wang algorithm for our actual research. Using the program, developed by Professor Swendsen, Wang, and Shafiqur Rahman, we conclude that there is no phase transition at a temperature above zero for this system. The above hypothesis was supported by three main observations. The first deals with the peak value of the specific heat as a function of temperature, which should diverge at the critical temperature for a phase transition. We found no such divergence for the five-state Potts system. Next, we used the Variable Sublattice Method, (VSM) developed by Professor Rahman, to examine the concentration of each type of spin as a function of temperature. Unlike the three- and four-state system, where a phase transitions do exist, there was no anomalous behavior of the spin concentration for the five-state system. Finally also using the VSM, the spin concentrations were plotted against the lattice size of the system for zero temperature. Thereby, we could extrapolate the concentration values for infinitely larger systems. These extrapolated concentration values turned out to be the same for every spin type, at all temperatures, leading to the conclusion that the system is disordered even at absolute zero. Thus, we conclude that the five-state antiferromagnetic Potts model has no phase transition above absolute zero.
Efficient Use of Monte Carlo Simulation Data Through the Use of the Single Histogram Method
Charles Ruggiero, 2002
The single-histogram method for optimized Monte Carlo data analysis will be presented in this paper. The effectiveness of the single histogram method in the study of phase transitions will be discussed. The method will then be applied to the Ising Model in two-dimensions and compared to results produced from multiple Monte Carlo simulations. The application of the single-histogram method will be discussed as it applies to the study of large systems used for simulation of phase transitions.
Using the Monte Carlo and Single Histogram Methods to Study the Energy and Magnetization of Complex Systems
Robyn Nelson, 2001
Since it is difficult, and in most cases impossible, to solve a complex ferromagnetic or antiferromagnetic system analytically, one must use computer simulation methods in order to study such a system. Various computer simulation methods have been developed to study such complex systems. In order to test these methods, they are used on simpler systems that can be solved analytically. This serves as a check so increasingly complex systems may be studied with greater confidence in the findings. One such Method is the Monte Carlo Method which can be used to gain information for the creation of a single histogram. The single histogram method is then used to create other histograms at different temperatures near the original temperature at which the data was collected.